Instantons in QM & Resurgent Expansions_zinn Justin

8/10/2019 Instantons in QM & Resurgent Expansions_zinn Justin

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arXiv:h

ep-ph/0405279v3

1Jul2004

Instantons in Quantum Mechanics

and Resurgent Expansions

Ulrich D. Jentschura

Physikalisches Institut der Universitat Freiburg,HermannHerderStrae 3, 79104 Freiburg im Breisgau, Germany and

National Institute of Standards and Technology,Gaithersburg, MD20899-8401, Maryland, USA

electronic mail address: [email protected].

Jean ZinnJustin

DAPNIA/DSM (Department dastrophysique, de physique des particules,de physique n ucleaire et de linst rumentation associee),

Commissariat a l Energie Atomique, Centre de Saclay, F-91191 Gif-sur-Yvette, France andInstitut de Mathematiques de JussieuChevaleret, Universite de Paris VII, France

electronic mail address: [email protected].

Abstract

Certain quantum mechanical potentials give rise to a vanishing perturbation seriesfor at least one energy level (which as we here assume is the ground state), but thetrue ground-state energy is positive. We show here that in a typical case, the eigen-value may be expressed in terms of a generalized perturbative expansion (resurgent

expansion). Modified BohrSommerfeld quantization conditions lead to generalizedperturbative expansions which may be expressed in terms of nonanalytic factorsof the form exp(a/g), where a > 0 is the instanton action, and power series inthe couplingg, as well as logarithmic factors. The ground-state energy, for the spe-cific Hamiltonians, is shown to be dominated by instanton effects, and we providenumerical evidence for the validity of the related conjectures.

Key words: General properties of perturbation theory; Asymptotic problems andpropertiesPACS: 11.15.Bt, 11.10.Jj

Preprint submitted to Elsevier Science 1 February 2008
http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3http://arxiv.org/abs/hep-ph/0405279v3


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Introduction.A number of intriguing and rather subtle issues are connectedwith simple RayleighSchrodinger perturbation theory when it is applied tocertain classes of one-dimensional quantum mechanical model problems, whichgive rise to divergent perturbation series and allow for the presence of instan-ton effects[1]. Of particular interest is the case of the symmetric double-well

potential[2,3]

Vdw(g, q) =1

2q2 (1g q)2 , (1)

the Hamiltonian being Hdw =1/2 (d/dq)2 + Vdw(g, q). There are severalpoints to note: (i) The perturbation series can be shown to be non-Borelsummable [2, 3] for positive g. (ii) The parity operation q 1 q leavesVdw(g,q/

g) invariant, and eigenfunctions are classified according to a prin-

cipal quantum number Nand the parity eigenvalue =. States with thesame principal quantum number but opposite parity are described by the sameperturbative expansion. (iii) The energy splitting between states of opposite

parity is described by nonanalytic factors of the form exp[1/(6g)]. In general,quantum tunneling may generate additional contributions to eigenvalues of or-der exp(const. /g), which have to be added to the perturbative expansion (fora review and more detail about barrier penetration in the semi-classical limitsee for example [4]). Dominant contributions to the Euclidean path integralare generated by classical configurations (trajectories) that describe quantummechanical tunneling among the two degenerate minima; their Euclidean ac-tion remains finite in the limit of large positive and large negative imaginarytime (for a review see [5]).

Thus, the determination of eigenvalues starting from their expansion for small

g is a non-trivial problem. Conjectures [6,7,8,9,10] have been discussed inthe literature which give a systematic procedure to calculate eigenvalues, forfiniteg , from expansions which are shown to contain powers of the quantitiesg, ln g and exp(const. /g), i.e. resurgent[11,12] expansions. Moreover, gen-eralized BohrSommerfeld formulae (see e.g. [13, Eq. (2)]) can be extractedby suitable transformations from the corresponding WKB expansions. (Thequantization conditions may also be derived, approximatively, from an exactevaluation of path integral in the limit of a vanishing instanton interaction,by taking into account an arbitrary number of tunnelings between the minimaof the potential [14,15,16].) Note that the relation to the WKB expansion isnot completely trivial. Indeed, the perturbative expansion corresponds (fromthe point of view of a semi-classical approximation) to a situation with con-fluent singularities and thus, for example, the WKB expressions for barrierpenetration are not uniform when the energy goes to zero.

Here, we are concerned with a modification of the double-well problem,

VFP(g, q) =1

2q2 (1g q)2 +g q 1

2, (2)

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the Hamiltonian being HFP = 1/2 (d/dq)2 + VFP(g, q). The potentialVFP(g, q) also contains a linear symmetry-breaking term. There are the fol-lowing points to note with regard toVFP(g, q):(i)parity is not conserved, andthere is no degeneracy of the spectrum on the level of the perturbative expan-sion. (ii) The perturbation series for the ground state vanishes identically to

all orders in the couplingg [17].(iii)The true ground-state energy is positive;in [17] it was shown that it fulfills 0 < E0 < Cexp(D/g), where C and Dare positive constants. Here, we present a resurgent expansion which naturallyleads to a generalization of perturbation theory valid for problematic poten-tials such asVFP(g, q). Furthermore, we conjecture that a complete descriptionof the energy eigenvalues can be obtained via a generalized BohrSommerfeldquantization condition which allows for the presence of nonanalytic contribu-tions of order exp[1/(3g)] for the ground state and of order exp[1/(6g)] forexcited states, and we present numerical evidence for the validity of this conjec-ture. We thereby attempt to provide a complete description of the eigenvaluesof the FokkerPlanck potential by a generalized perturbation series involving

instanton contributions. More general cases are treated in [15,16].

We are not concerned here with supersymmetric quantum mechanics. In thiscontext, the FokkerPlanck Hamiltonian has received some attention in thepast two decades (see e.g. [18,19]). Instead, we rather attempt to find the suit-able generalization of perturbation theory that gives us an exact generalizedsecular equation for the energy eigenvalues which in turn yields a general-ization of perturbation theory suitable to the problem at hand. We will notsatisfy ourselves with an approximate solution of the problem but we attemptto find complete expressions for the energy eigenvalues in terms of resurgentexpansions.

FokkerPlanck Hamiltonian.The particularly interesting HamiltonianVFP(g, q) has been studied in [17]. The spectra of the Hamiltonians Hdw andHFPare invariant under the scale transformation q q/gand can thereforebe written alternatively as

Hdw=g2

d

dq

2+

1

gVdw(q), (3a)

Vdw(q) =

1

2q

2

(1 q)2

, (3b)

HFP =g2

d

dq

2+

1

gVFP(q), (3c)

VFP(q) = Vdw(q) + g

q 12

. (3d)

This is a representation which illustrates that g takes the formal role of h

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and that the linear symmetry-breaking term in VFP(q) in fact represents anexplicit correction to the potential of relative order g .

For the double-well potential, the following two functions enter into the gen-eralized BohrSommerfeld quantization formula[9,13,14],

Bdw(E, g) = E+ g

3E2 +1

4

+ g2

35E3 +

25

4E

+O

g3

, (4a)

Adw(E, g) = 1

3g+ g

17E2 +

19

12

+ g2

227 E3 +

187

4 E

+O

g3

. (4b)

The quantization condition and the resurgent expansion for the eigenvaluesread

12

1

2Bdw(E, g)

2

g

Bdw(E,g)exp

Adw(E, g)

2

= i , (5)

and

E,N(g) =l=0

E(0)N,l g

l

+n=1

2

g

Nn e

1/6g

g

n n1k=0

ln

2

g

k l=0

eN,nkl gl. (6)

Here, theE(0)

N,l

are perturbative coefficients[6], and the expression forE,N(g)follows naturally from an expansion of (5) in powers of g, ln(g), andexp[1/(6g)]. The index n characterizes the order of the instanton contri-bution (n = 1 is a one-instanton, etc.). The conjecture (5) has been verifiednumerically to high accuracy[13].

Insight can be gained into the problem by considering the logarithmic deriva-tiveS(q) = g (q)/(q), which for a general potentialVsatisfies the Riccatiequation

g S(q) S2(q) + 2V(q) 2gE= 0 . (7)This equation formally allows for solution withE= 0 (and implies a vanishing

perturbation series), if the potential V(q) has the following structure:

V(q) =1

2

U2(q) g U(q)

. (8)

Indeed, a formal solution ofH = 0 in this case is given by

(q) = exp

1

g

qdqU(q)

. (9)

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The Hamiltonian VFP is of this structure, with U(q) = UFP(q) = q(1 q).This fact leads to the peculiar properties ofVFP, and indeed the Hamiltoniansdiscussed in[17] belong to this class. The intriguing questions raised by theremarks made in [17] find a natural explanation in terms of generalized BohrSommerfeld quantization conditions, and resurgent expansions.

Before discussing VFP, we first make a slight detour and consider the specialcase UII(q) = q

3 +q. The potential 12U2II(q) has no degenerate minima, and

thus there are no instantons to consider. Indeed, in the case of the HamiltonianHII = (g/2)(d/dq)2 + [U2II(q)g UII(q)]/(2g) (we follow the notation of[17]),the expression (9) may be utilized for the construction of a normalizable eigen-function of the Hamiltonian which reads II(q) = exp[(q2/2 + q4/4)/g] andhas the eigenvalue E= 0.

In the case of the potential UFP(q) =q(1 q), the issue is more complicatedbecause the wave function

(q) = exp

1

g

q3

3 q

2

2

(10)

is not normalizable, and thus is not an eigenfunction. An analogy of the Riccatiequation (7) with the FokkerPlanck equation suggests that the case E = 0be identified with an equilibrium probability distribution. Therefore, the non-normalizable wave function (10) may naturally be identified with a pseudo-equilibrium distribution.

Instanton action.The Euclidean instanton action for the ground state of the

FokkerPlanck potential is given by [8,9]

a= 2 10

dq UFP(q) = 2 16

=1

3, (11)

and it is this quantity which determines the leading contribution to the ground-state energy of order exp[1/(3g)]. We conjecture here the following general-ized quantization condition for the eigenvalues of the Hamiltonian (3c)

1

(BFP(E, g)) (1BFP(E, g))+

2

g

2BFP(E,g) exp(AFP(E, g))2

= 0 .

(12)This condition is different from what would be obtained if one were to considerperturbation theory alone. Indeed, the perturbative quantization conditionreads

BFP(E, g) = N , (13)

with integer N 0. The functions BFP and AFP determine the perturbativeexpansion, and the perturbative expansion about the instantons, in higherorder. They have the following expansions,

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BFP(E, g) =E+ 3 E2 g+

35 E3 +

5

2E

g2

+

1155

2 E4 + 105 E2

g3 +

45045

4 E5 +

15015

4 E3 +

1155

8 E

g4

+969969

4

E6 +255255

2

E4 +111111

8

E2 g5+

22309287

4 E7 +

33948915

8 E5 +

3556553

4 E3 +

425425

16 E

g6 ,

+

2151252675

16 E8 +

557732175

4 E6

+ 379257879

8 E4 + 4157010 E2

g7 +O(g8) , (14a)

AFP(E, g) = 1

3g+

17 E2 +

5

6

g +

227 E3 +

55

2 E

g2

+4743112

E4 +11485

12 E2 +

1105

72 g3

+

317629

4 E5 +

64535

2 E3 +

4109

2 E

g4

+

26145967

15 E6 +

25643695

24 E4 +

4565723

30 E2 +

82825

48

g5

+

812725953

20 E7 +

280162805

8 E5

+1057433447

120 E3 +

20613005

48 E

g6 +O(g7) . (14b)

The calculation of the functions A- and B-functions, for general classes of

potentials, is described in more detail in [15, 16]. On the basis of (13) and(14a), we obtain the following perturbative expansion E

(pert)N (g) up to and

including terms of order g3, for general N,

E(pert)N (g) N3 N2 g

17 N3 +

5

2N

g2

375

2 N4 +

165

2 N2

g3+O(g4) .

(15)Here, the upper index (0) means that only the perturbative expansion (inpowers ofg) is taken into account. For the ground state (N= 0), all the termsvanish, whereas for excited states with N = 1, 2, . . . , the perturbation seriesis manifestly nonvanishing.

The quantization condition (12) is conjectured to be the secular equationwhose solutions determine the the energy eigenvalues of the FokkerPlanck po-tential (3d). The Eqs. (14a) and (14b) can be used to expand the ground-stateenergy eigenvalue up to sixth order in the nonperturbative factor exp(1/3g),and up to seventh order in the coupling g . The general structure of the resur-gent expansions determined by (12) differs slightly for the ground state incomparison to the excited states. This will be shown below, with a special

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emphasis on the ground state.

Resurgent expansion for the ground state.Based on (12), we derive the fol-lowing expansion for the ground-state energy (N = 0) of the FokkerPlanckpotential (3d):

E(0)FP(g) =

n=1

e1/3g

2

n n1k=0

ln

2

g

k l=0

f(0)nkl g

l. (16)

For small coupling g, this expansion is strongly dominated by the one-instanton effect (n= 1). An explicit calculation using (14a) and (14b) leads tothe following expansion for the ground-state energy of the Hamiltonian HFP,

which is valid up to terms of order [exp(1/3g)]2

,

E(0)FP(g)

exp 1

3g

2

1 5

6g 155

72 g2 17315

1296 g3 3924815

31104 g4

392481531104

g4 294332125186624

g5 1639682311756718464

g6

1812431458772540310784

g7 185875465098807251934917632

g8 +O(g9)

+O

[exp(1/3g)]2

. (17)

Because the perturbation series (15) vanishes forN= 0, the resurgent expan-sion starts with the one-instanton effect. Indeed, Eq. (17) is the one-instantoncontribution to the energy, characterized by a nonanalytic factor exp(1/3g)which is multiplied by a (divergent, nonalternating) power series in g. Thisnonalternating series in g may be resummed by a generalized Borel method(the generalized Borel sum finds a natural representation in the sense of dis-tributional Borel summability, which is effectively a Borel sum in complex

directions of the parameters, see e.g. [20,21,22,23,24]).

In analogy to the double-well potential, the imaginary part which is gener-ated by this procedure (the discontinuity of the distributional Borel sumin the terminology of [22]) is compensated by an explicit imaginary part thatstems from the two-instanton effect. We supplement here the first few termsof the two-instanton shift of the ground-state eigenvalue [terms with n = 2 inEq. (16)]:

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[exp(1/3g)]2(2)2

2 ln

2

g

+ 2

+g

10

3 ln

2

g

10

3 3

+O(g2 ln g)

. (18)

Here, = 0.577216 . . . is Eulers constant.

The perturbative coefficients about one instanton, called f(0)10K in Eq. (16),

grow factorially as

f(0)10K

3K(K)

, K . (19)

It is an easy exercise to verify that this factorial growth exactly leads to animaginary part that is canceled by the imaginary part that results from theanalytic continuation of the expression 2 ln(

2/g)+2in (18) from negative to

positiveg . The explicit coefficients in (17) are consistent with the asymptoticformula (19).

We have performed extensive numerical checks on the validity of the expansion(17). For example, at g= 0.007, the ground-state energy, obtained numerically,is

E(0)FP(0.007) = 3.300 209 301 936(1) 1022 (20)

based on a calculation with a basis set composed of up to 300 harmonic os-cillator eigenstates. The numerical uncertainty is estimated on the basis ofthe apparent convergence of the results under an appropriate increase of the

number of states in the basis set.

When adding all terms up to the order ofg9 in the perturbative expansionabout the leading instanton (the first eight terms are given in Eq. (17), furtherterms are available for download [25]), we obtain

E(0)FP(0.007) 3.300209301942 1022 . (21)

With the term of order g10 included, we have

E(0)FP(0.007) 3.300209301936 1022 (22)

in full agreement with (20) to all decimals shown.

We should clarify why then-instanton contribution in the resurgent expansion(6) for the double-well potential (3b) involves the nth power of the expres-sion exp(1/6g), while in the case of the ground-state of the FokkerPlanckHamiltonian it involves the nth power of exp(1/3g). One may answer thisquestion by observing that in a symmetric potential, instanton configurationswith an odd number of tunnelings between the minima yield a nonvanishing

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contribution to the path integral, and therefore, the one-instanton config-uration in the double-well is a trajectory that starts in one well and ends inthe other. The linear symmetry-breaking term of the FokkerPlanck potentiallifts this degeneracy; the leading, one-instanton shift of the ground state isnow a configuration in which the particle returns to the well from which it

started; the instanton action is therefore twice as large and the two-instantoncontribution (the bounce-configuration in the case of the double-well poten-tial) becomes the one-instanton solution in the case of the ground-state of theFokkerPlanck equation.

As a last remark, it is useful to observe that although the correction termg(q 1/2) in Eq. (3d) vanishes in the limit g 0, one cannot recover thedouble-well quantization condition (5) from (12) in this limit; it is nonuniform.

Resurgent expansion for excited states.The energy of excited states (N >0)is dominated, for small g, by the perturbative expansion (15) which is mani-festly nonvanishing to all orders ing . Because the symmetry is broken only atorderg [see Eq. (3d)], and because the dominance of the perturbation series(expansion in g) is fully restored for excited states, the resurgent expansioninduced by (12) becomes very close to the analogous expansion for the statesof the double-well potential (6). By direct expansion of (12), taking advantageof the the functional form of the dependence ofAFP(g) and BFP(g) on g , weobtain the resurgent expansion

E(,N>0)FP (g) =E

(pert)N (g) +

n=1

[ N(g)]nn1k=0

ln

2

g

k l=0

f(N)nkl g

l. (23)

Here, = is a remnant of the parity which is broken by VFP(g), but onlyat order g, E

(pert)N (g) is the perturbative expansion given in (15), and N(g)

is given by

N(g) =

2

2N1 exp(1/6g)gN

N! (N 1)!

. (24)

For completeness, we indicate here the first few terms for the resurgent ex-pansion of the states withN= 1, but opposite (perturbatively broken) parity= , to leading order in the coupling up to the three-instanton term,

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E(,N=1)FP (g) = 1 +O(g) 1(g)(1 +O(g))

+ [1(g)]2

ln

2

g

+ 1

2+ O(g ln g)

[1(g)]3

32

ln22

g

+3

2+ 3

ln

2

g

+5

8 3

2+

3

22 +

2

12+ O(g ln g)

. (25)

Conclusions.We have presented the quantization condition (12) which, to-gether with Eqs. (14a) and (14b), determines the resurgent expansions for anarbitrary state (quantum number N) of the FokkerPlanck potential (3c) upto seventh order in the coupling g, and up to and including the six-instantonorder. For general N >0, the perturbation series is nonvanishing [see Eq. (15)],and the instanton contributions, for small coupling, yield tiny corrections tothe energy. However, for the ground state withN= 0, the perturbation seriesvanishes to all orders in the coupling, and the resurgent expansion (16) for theground-state energy of the FokkerPlanck potential (3c) is dominated by thenonperturbative factor exp(1/3g) that characterizes the one-instanton con-tribution to the ground-state energy. The nonperturbative factor exp(1/3g)is multiplied by a factorially divergent series [see Eqs. (17) and (19)]; this isthe natural structure of a resurgent expansion which holds also for the double-well potential [see Eq. (6)]. The basic features of this intriguing phenomenon

have been described in [17]; they find a natural and complete explanation interms of the resurgent expansions discussed here.

Concepts discussed in the current Letter may easily be generalized to moregeneral symmetric potentials with degenerate minima, potentials with twoequal minima but asymmetric wells, and periodic-cosine potentials (some fur-ther examples are discussed in [15,16]). There is a well-known analogy betweena one-dimensional field theory and one-dimensional quantum mechanics, theone-dimensional field configurations being associated with the classical trajec-tory of the particle. Indeed, the loop expansion in field theory corresponds tothe semi-classical expansion [26, chapter 6]. Therefore, one might hope that

suitable generalizations of the methods discussed here could result in new con-jectures for problems where our present understanding is (even) more limited.

Resurgent expansions appear to be of wide applicability in a number of caseswhere ordinary perturbation theory, even if augmented by resummation pre-scriptions, fails to described physical observables such as energy eigenvalueseven qualitatively. This has been demonstrated here using the FokkerPlanckpotential as an example.

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Acknowledgments. The authors would like to acknowledge the Institute ofPhysics, University of Heidelberg, for the stimulating atmosphere during a visitin January 2004, on the occasion of which part of this work was completed, andthe AlexandervonHumboldt Foundation for support. Roland Rosenfelderand Peter Mohr are gratefully acknowledged for helpgul conversations. The

stimulating atmosphere at the National Institute of Standards and Technologyhas contributed to the completion of this project.

References

[1] J. C. LeGuillou and J. Zinn-Justin, Large-Order Behaviour of Perturba-tion Theory (North-Holland, Amsterdam, 1990).

[2] E. Brezin, G. Parisi, and J. Zinn-Justin, Phys. Rev. D 16, 408 (1977).[3] E. B. Bogomolny, Phys. Lett. B91, 431 (1980).

[4] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed.(Clarendon Press, Oxford, 2002), ch. 43.

[5] H. Forkel, A Primer on Instantons, e-printhep-ph/0009136.[6] J. Zinn-Justin, J. Math. Phys. 22, 511 (1981).[7] J. Zinn-Justin, Nucl. Phys. B192, 125 (1981).[8] J. Zinn-Justin, Nucl. Phys. B218, 333 (1983).[9] J. Zinn-Justin, J. Math. Phys. 25, 549 (1984).[10] Several results have also been reported in J. Zinn-Justin, contribution to

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de Monvel ed., Collection Travaux en cours, 47, Hermann (Paris 1994).[11] J. Ecalle, Les Fonctions Resurgentes, Tomes IIII, PublicationsMathematiques dOrsay, France (19811985).

[12] M. Stingl, FieldTheory Amplitudes as Resurgent Functions, e-printhep-ph/0207049.

[13] U. D. Jentschura and J. Zinn-Justin, J. Phys. A 34, L253 (2001).[14] L. Boutet de Monvel (Ed.), Methodes Resurgentes (Hermann, Paris,

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sults I: Conjectures, WKB Expansions, and Instanton Interactions, Ann.Phys. (N.Y.), at press (2004).

[16] J. Zinn-Justin and U. D. Jentschura, MultiInstantons and Exact Re-sults II: Specific Cases, Higher-Order Effects, and Numerical Calculations,Ann. Phys. (N.Y.), at press (2004).

[17] I. W. Herbst and B. Simon, Phys. Lett. B 78, 304 (1978).[18] E. Witten, Nucl. Phys. B 185, 513 (1981).[19] H. Aoyama, H. Kukuchi, I. Okuochi, M. Sato, and S. Wada, Nucl. Phys.

B 533, 644 (1999).[20] I. W. Herbst and B. Simon, Phys. Rev. Lett. 41, 67 (1978).

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[21] V. Franceschini, V. Grecchi, and H. J. Silverstone, Phys. Rev. A32, 1338(1985).

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[24] U. D. Jentschura, Phys. Rev. D 62, 076001 (2000).[25] See the URL http://tqd1.physik.uni-freiburg.de/~ulj.[26] C. Itzykson and J. B. Zuber,Quantum Field Theory(McGraw-Hill, New

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